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\title{Spin Precession in a Magnetic Field}
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\author{Your Name}

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In this article, we examine the precession of a spin $\frac{1}{2}$ in a magnetic field.

We consider a magnetic field $\vb{B} = B\vu{z}$. The Hamiltonian is $H = \vb{H} = -\vb{\mu}\vdot\vb{B} = -\gamma\vb{B}\vdot\vb{S}$. For a spin $\frac{1}{2}$, $H = -\frac{\gamma\hbar}{2}\vb{B}\vdot\vb{\sigma}$. Let $\omega = \gamma B$, where $\vb{B} = \vu{n}$. Choosing units such that $\hbar = 1$, we have $H = -\frac{\omega}{2} \vu{n}\vdot\vb{\sigma}$.

We measure time in units of $\frac{1}{\omega}$. Therefore one oscillation corresponds to $t = \pi$. In these units, the Hamiltonian is simply $H = - \vu{n}\vdot\vb{\sigma}$. Choosing the magnetic field to be along the $z$ direction, that is $\vb{B} = B\vu{z}$, the Hamiltonian is $H = -\sigma_z$.

Figure \ref{fig:expsigma} shows the time evolution of the expectation value of the Pauli operators $\sigma_x, \sigma_y, \sigma_z$. As we expect since $\sigma_z$ commutes with the Hamiltonian, the expectation value $\expval{\sigma_z}$ does not change with time. Figure \ref{fig:blochvector} shows a few snapshots in the precession of the spin on the Bloch sphere during one oscillation. As expected we see a clockwise precession of the spin and the Bloch vector.

\begin{figure}
  \centering
  \includegraphics{exp_sigma.eps}
  \caption{The expectation values of the Pauli operators $\expval{\sigma_x}, \expval{\sigma_y}, \expval{\sigma_z}$ under the Hamiltonian as a function of time. Note that the expectation value $\expval{\sigma_z}$ does not change as $\sigma_z$ commutes with the Hamiltonian.}
  \label{fig:expsigma}
\end{figure}



\begin{figure}
  \centering
  \includegraphics{bloch.eps}
  \caption{The Bloch vector during one cycle of precession about the magnetic field. The intial Bloch vector is marked with a blue circle and the next Bloch vector is marked with a red square for reference of position and direction of motion.}
  \label{fig:blochvector}
\end{figure}


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